Problem 72 - University Physics with Modern Physics

Question

Electron Spin Resonance. Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state. (a) Find the magnetic-field magnitude \(B\) required for this transition in a hydrogen atom with \(n=1\) and \(l=0\) to be induced by microwaves with wavelength \(\lambda\) . (b) Calculate the value of \(B\) for a wavelength of 3.50 \(\mathrm{cm} .\)

Step-by-Step Solution

Answer

The magnetic field magnitude required is approximately 0.327 T.

Step 1: Understand the Electron Spin Transition

Electrons have two spin states: up and down. The energy difference between these states can be defined as \(\Delta E = g \mu_B B\), where \(g\) is the g-factor of the electron (approximately 2), \(\mu_B\) is the Bohr magneton, and \(B\) is the magnetic field strength. For a transition to occur, this energy difference must match the energy of the absorbed photon.

Step 2: Relate Energy to Wavelength

The energy of a photon is given by \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the photon. For resonance, this matches the energy difference \(\Delta E = g \mu_B B\).

Step 3: Set Up the Equation for B

Equate the energy of the photon to the energy difference due to the magnetic field: \(\frac{hc}{\lambda} = g \mu_B B\). Solve for \(B\), which gives \(B = \frac{hc}{g \mu_B \lambda}\).

Step 4: Input Known Constants

Substitute the constants into the equation: - Planck's constant \(h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s}\)- Speed of light \(c = 3 \times 10^8 \text{ m/s}\)- Bohr magneton \(\mu_B = 9.274 \times 10^{-24} \text{ J/T}\)- g-factor \(g = 2\). Adjust \(\lambda\) to meters: \(3.50 \text{ cm} = 0.0350 \text{ m}\).

Step 5: Calculate \(B\) for Given Wavelength

Substitute into the equation:\[ B = \frac{(6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s})(3 \times 10^8 \text{ m/s})}{2 (9.274 \times 10^{-24} \text{ J/T}) (0.0350 \text{ m})} \]This calculation yields:\[ B \approx 0.327 \text{ T} \]

Step 6: Conclusion

The required magnetic field magnitude \(B\) for the electron spin transition to occur at a wavelength of 3.50 cm is approximately 0.327 tesla.

Key Concepts

Magnetic Field

When it comes to electron spin resonance, the magnetic field plays a crucial role. The magnetic field, denoted as \( B \), is the external force that influences the spin states of electrons. In simple terms, you can think of the magnetic field as a sort of invisible force that impacts charged particles, such as electrons. In the context of electron spin resonance, the magnetic field is necessary to create a difference in energy levels between the electron's spin states. This energy difference enables electrons to transition from one spin state to another by absorbing a photon.

• Magnetic fields vary in strength and can be measured in units called tesla (T).
• The magnetic field strength required for a specific spin transition is determined by the properties of the electron and the wavelength of the absorbed photon.

For resonance to occur, the magnetic field creates an energy gap that matches the energy of incoming photons, thus enabling a transition between spin states.

Photon Energy

Photon energy is the key link between the magnetic field and electron spin state transitions. A photon is a particle of light, and its energy is what enables an electron within a magnetic field to change its spin state. To determine the energy of a photon, we use the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}) \), \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength of the photon. This dependency establishes a direct relationship between photon energy and wavelength.

• Photon energy increases as the wavelength decreases.
• The energy of a photon must match the energy difference created by the magnetic field for a spin transition to occur.

Understanding photon energy helps explain how specific wavelengths facilitate electron transitions within a magnetic field for electron spin resonance.

Hydrogen Atom

The hydrogen atom is one of the simplest atomic models, consisting of one electron orbiting a single proton. This simplicity makes it an ideal atom for studying basic quantum mechanical processes, like electron spin resonance.In electron spin resonance studies involving hydrogen atoms, only the ground state or lowest energy level, also known as \( n=1 \), is typically considered. At this state, the single electron within a hydrogen atom can occupy different spin states influenced by an external magnetic field.

• Hydrogen's simplicity aids in isolating the effects of magnetic fields on electron spins.
• Observing spin transitions in a hydrogen atom can serve as a base model for understanding more complex atoms and molecules.

Through its single electron, the hydrogen atom provides foundational insights into the dynamics of quantum mechanics and electron spin states.

Spin States

Spin states refer to the intrinsic angular momentum of an electron, often visualized as 'up' or 'down' based on its orientation relative to an external magnetic field. Even though the concept of spin may seem abstract, it plays a critical role in electron spin resonance.For electrons:

• There are typically two spin states: +1/2 (spin-up) and -1/2 (spin-down).
• The energy levels of these states split in an external magnetic field, creating an energy difference that can be quantified with the formula \( \Delta E = g \mu_B B \), where \( \Delta E \) is the energy difference, \( g \) is the electron g-factor (approximately 2), and \( \mu_B \) is the Bohr magneton.

Electron spin resonance occurs when a photon of the right energy matches this energy difference, allowing the electron to transition from one spin state to another. This transition is fundamental to understanding quantum mechanics and is utilized in various spectroscopic techniques.